with g⁢(X)gXg(X) and h⁢(X)hXh(X) certain polynomials in D⁢[X]DXD[X]. From these equations one infers that f⁢(X)fXf(X) is a constant polynomial ccc and h⁢(X)hXh(X) is a first degree polynomial b0+b1⁢Xsubscriptb0subscriptb1Xb_{0}\!+\!b_{1}X (b1≠0subscriptb10b_{1}\neq 0). Thus we obtain the equation

c⁢b0+c⁢b1⁢X=X,csubscriptb0csubscriptb1XXcb_{0}+cb_{1}X\;=\;X,

which shows that c⁢b1csubscriptb1cb_{1} is the unity 1 of DDD. Thus c=f⁢(X)cfXc=f(X) is a unit of DDD, whence

where u⁢(X),v⁢(X)∈D⁢[X]uXvXDXu(X),\,v(X)\in D[X]. This equation cannot be possible without that aaa times the constant term of u⁢(X)uXu(X) is the unity. Accordingly, aaa has a multiplicative inverse in DDD. Because aaa was arbitrary non-zero elenent of the integral domain DDD, DDD is a field.